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13
Composition of random transpositions
 Department of Mathematics University of British Columbia
, 2005
"... Let Y = (y1,y2,...), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,...,n}. We prove that if c> 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the PoissonDirichlet distribution with paramenter 1, where t ..."
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Let Y = (y1,y2,...), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,...,n}. We prove that if c> 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the PoissonDirichlet distribution with paramenter 1, where the function f is known explicitly. A new proof is presented of the theorem by Diaconis, MayerWolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulationfragmentation process. 1
Probabilistic transforms for combinatorial urn models
 Combin. Probab. Comput
, 2002
"... In this paper, we present several probabilistic transforms related to classical urn models. These transforms render the dependent random variables describing the urn occupancies into independent random variables with appropriate distributions. This simplifies the analysis of a large number of proble ..."
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Cited by 9 (1 self)
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In this paper, we present several probabilistic transforms related to classical urn models. These transforms render the dependent random variables describing the urn occupancies into independent random variables with appropriate distributions. This simplifies the analysis of a large number of problems for which a function under investigation depends on the urn occupancies. The approach used for constructing the transforms involves generating functions of combinatorial numbers. We also show, by using Tauberian theorems derived in this paper, that under certain conditions the asymptotic expressions of target functions in the transform domain are identical to the asymptotic expressions in the inverse–transform domain. Therefore, asymptotic information about certain statistics can be gained without evaluating the inverse transform. 1
Convergence to the Coalescent with Simultaneous Multiple Mergers
, 2002
"... The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized in [13] in terms of a sequence of measures de ned on the nitedimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure ..."
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The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized in [13] in terms of a sequence of measures de ned on the nitedimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure de ned on the in nite simplex was suggested in [17].
Thermodynamic limits of macroeconomic or financial models: Oneand twoparameter PoissonDirichlet models
 Journal of Economic Dynamics and Control
"... This paper examines asymptotic behavior of two types of economic or financial models with many interacting heterogeneous agents. They are oneparameter PoissonDirichlet models, also called Ewens models, and its extension to twoparameter PoissonDirichlet models. The total number of clusters, and ..."
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This paper examines asymptotic behavior of two types of economic or financial models with many interacting heterogeneous agents. They are oneparameter PoissonDirichlet models, also called Ewens models, and its extension to twoparameter PoissonDirichlet models. The total number of clusters, and the components of partition vectors (the number of clusters of specified sizes), both suitably normalized by some powers of model sizes, of these classes of models are shown to be related to the MittagLeffler distributions. Their behavior as the model sizes tend to infinity (thermodynamic limits) are qualitatively very different. In the oneparameter models, the number of clusters, and components of partition vectors are both selfaveraging, that is, their coefficients of variations tend to zero as the model sizes become very large, while in the twoparameter models they are not selfaveraging, that is, their coefficients of variations do not tend to zero as model sizes becomes large. KeyWords:PoissonDirichlet distributions; MittagLeffler distributions; Thermodynamic limits; Nonself averaging phenomena, Power laws. ∗The author is grateful for many helps he received from M. Sibuya. Comments by Ted Theodosopoulos, Thomas Lux and an anonymous referee’s comments have been useful in revising the original draft. I thank them all. 1
Sampling formulae arising from random Dirichlet populations
, 2004
"... Consider the random Dirichlet partition of the interval into n fragments at temperature θ> 0. Some statistical features of this random discrete distribution are recalled, together with explicit results on the law of its sizebiased permutation. Using these, preasymptotic versions of the Ewens an ..."
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Consider the random Dirichlet partition of the interval into n fragments at temperature θ> 0. Some statistical features of this random discrete distribution are recalled, together with explicit results on the law of its sizebiased permutation. Using these, preasymptotic versions of the Ewens and DonnellyTavaréGriffiths sampling formulae from finite Dirichlet partitions are computed exactly. From these, new proofs of the usual sampling formulae from random proportions with GEM(γ) distribution are supplied, when considering the Kingman limit n ↑ ∞, θ ↓ 0 1 while nθ = γ> 0.
Random Partitioning Problems Involving Poisson Point Processes On The Interval
, 2004
"... Suppose some random resource (energy, mass or space) χ ≥ 0 is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume Eχ = θ < ∞ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can natural ..."
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Suppose some random resource (energy, mass or space) χ ≥ 0 is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume Eχ = θ < ∞ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can naturally be identified with the study of infinitely divisible random variables with Lévy measure concentrated on the interval. Special emphasis is put on these special partitioning models in the PoissonKingman class. The masses attached to the atoms of such partitions are sorted in decreasing order. Considering nearestneighbors spacings yields a partition of unity which also deserves special interest. For such partition models, various statistical questions are addressed among which: correlation structure, cumulative energy of the first K largest items, partition function, threshold and covering statistics, weighted partition, Rényi’s, typical and sizebiased fragments size. Several physical images are supplied. When the unbounded Lévy measure of χ is θx −1 ·I (x ∈ (0,1))dx, the spacings partition has GriffithsEngenMcCloskey or GEM(θ) distribution and χ follows Dickman distribution. The induced partition models have many remarkable peculiarities which are outlined. The case with finitely many (Poisson) fragments in the partition law is also briefly addressed. Here, the Lévy measure is bounded.
OCCUPANCY DISTRIBUTIONS ARISING IN SAMPLING FROM GIBBSPOISSON ABUNDANCE MODELS
, 2013
"... Abstract. Estimating the number n of unseen species from a k−sample displaying only p ≤ k distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with ..."
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Abstract. Estimating the number n of unseen species from a k−sample displaying only p ≤ k distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with GibbsPoisson distribution. A k−sample drawn from the n−species abundances vector is the one obtained while conditioning it on summing to k. We discuss the sampling formulae (species occupancy distributions, frequency of frequencies) in this context. We then develop some aspects of the estimation of n problem from the size k of the sample and the observed value of Pn,k, the number of distinct sampled species. It is shown that it always makes sense to study these occupancy problems from a GibbsPoisson abundance model in the context of a population with infinitely many species. From this extension, a parameter γ naturally appears, which is a measure of richness or diversity of species. We rederive the sampling formulae for a population with infinitely many species, together with the distribution of the number Pk of distinct sampled species. We investigate the estimation of γ problem from the sample size k and the observed value of Pk. We then exhibit a large special class of GibbsPoisson distributions having the property that sampling from a discrete abundance model may equivalently be viewed as a sampling problem from a random partition of unity, now in the continuum. When n is finite, this partition may be built upon normalizing n infinitely divisible iid positive random variables by its partial sum. It is shown that the sampling process in the continuum should generically be biased on the total length appearing in the latter normalization. A construction with sizebiased sampling from the ranked normalized jumps of a subordinator is also supplied, would the problem under study present infinitely many species. We illustrate our point of view with many examples, some of which being new ones.
Heterogeneous Agents: Asymptotic Behavior of One and TwoParameter PoissonDirichlet Distributions
, 2005
"... CIRJE Discussion Papers can be downloaded without charge from: ..."
Financial Models: One and TwoParameter PoissonDirichlet Models
, 2005
"... CIRJE Discussion Papers can be downloaded without charge from: ..."